Coadjoint-orbit conjecture for moduli of weighted prequantizable Lagrangian embeddings

Let $(M,\a0\omega)$ be a symplectic manifold and let $(S,\mu)$ be a weighted compact manifold. Consider the moduli space obtained from prequantizable Lagrangian embeddings of $S$ into $M$, modulo volume-preserving diffeomorphisms of $S$.

Coadjoint-orbit conjecture. The moduli space of weighted prequantizable Lagrangian embeddings is a symplectic manifold whose connected components are related to the coadjoint orbits of the symplectomorphism group.

For isotropic embeddings, related symplectic-quotient and coadjoint-orbit results are known under additional topological assumptions such as vanishing first cohomology of $S$; the statement without that assumption is presented as a direction for further study.